On the Instability Threshold of Journal Bearing Supported Rotors_VERY GOOD

8/9/2019 On the Instability Threshold of Journal Bearing Supported Rotors_VERY GOOD

1/18


2/18

International Journal o Rotating Machinery

once these requencies are not usually taken into account inthe classical analysis.

Aiming at a better estimation o a rotor critical speed,Stodola [ ] and Hummel [ ] considered the exibility o the bearing oil lm, introducing the idea o representingits dynamic properties by means o springs and dampers.

Tis idea was explored in a more ormal way in the work o Lund [ ], where equivalent linearcoefficients o stiffness anddamping were calculated through the simpli ed solution o the Reynolds equation. Recently, a nite-difference methodwas used to solve the Reynolds equation in the work o Machado and Cavalca [ ], and a nite-volume method wasused in Machado [ ], both intending to calculate moreaccurate equivalent coefficients, based on the previous work o Lund.

During the development o the equivalent linear coeffi-cients, other researches were made towards nonlinear meth-ods; or example, Capone [ , ] presented a nonlinearsolution or the Reynolds equation, obtaining a nonlinear

equation or the orce exerted by the journal bearings on theshaf.In this context, many authors [ ] used nite element

models (FEM) o shafs together with journal bearings mod-els to increase the understanding o complex rotors. Lima[ ] made simulations with Capones model and extendedit to tilting-pad bearings. Castro et al. [ ] used the samebearing model with a FEM model based in the work o Nelson and McVaugh [ ] to study the stability threshold o horizontal and vertical machines, achieving good agreementwith experimental results.

In the existing literature, oil-whirl is commonly re erredto as a weak sel -excited vibration, nevertheless, treated asinstability. Te rst part o this statement relies on the actthat oil-whirl effect is not observable in the rotor, onceits vibration is dominated by the unbalance orces. Te vibrationamplitudeo the shaf inside the bearings is a strong

unction o the eccentricity o the shaf within the bearingclearance, that is, the bearing eccentricity. An importantconcept regarding this matter is the eccentricity ratio, whichis a dimensionless quantity given by the bearing eccentricity divided by the bearing radial clearance. Concerning verticalrotors or lightly loaded horizontal rotors (with very low eccentricity ratio) the oil-whirl vibration in the bearing islimited by the uid orces and the bearing clearance; this

occurs because, as discussed later, in these rotors the cross-coupled coefficients o stiffness increase drastically as therotor accelerates, overcoming the dissipative orces o thedamping coefficients.

In some cases, this vibration due to oil-whirl may causeheating and wear in the bearing, justi ying the use o theterm instability . In such light rotors, it is also possible that theoil-whirl component becomes very small or vanishes duringthe rotor crossover through the critical speed, giving placeto the rise o the synchronous component due to unbalance[ ]. For horizontal rotors with higher eccentricity ratio, thesubsynchronous oil-whirl component may be present in thebearings but usually with small amplitudes.

Tere ore, when talking about stability threshold onhorizontal rotors, oil-whip is the phenomenon usually con-sidered. However, experimental evidences have shown that,in some cases, the instability threshold may occur soonerthan the expected oil-whip requency. Such observationswere made in a test rig at the Laboratory o Rotating

Machinery in the University o Campinas, when simulatinga procedure or stability identi cation o journal bearingsupported machines.

Tereby, the correct identi cation o the instability threshold is mandatory to ensure the expected per ormancein a secure operating condition, still during the design stage,remarkably in the case o high speed exible rotors (e.g.,multistage rotors present in power plants). Te previousknowledge o the instability threshold may be also use ulto the development o control systems to stabilize the uid-induced instabilities, allowing the rotors to sa ely operatein higher rotational speeds. Siqueira et al. [ ] developeda linear parameter varying (LPV) control, which takes intoaccount the parameter dependency o the system (in thiscase, the rotational speed), using the H-in nite technique. Asigni cantvibration amplitude reduction during the crossingo the critical speed in a run-up simulation was obtainedusing a magnetic actuator. Te authors stated that an impor-tant actor in the control design was a representative model;which they obtained using the linear coefficients approach.Riemann et al. [ ] took a step urther and managed tocontrol the oil-whip instability using a -synthesis control,extending the rotor operational range rom Hz (oil-whipthreshold) up to Hz in an experimental test rig; a magneticactuator was used as well. uma et al. [ ] extended theinstability threshold o a rigid rotor rom rpm to

rpm using two piezoactuators at the bearing bushingand a simple proportional controller.

In this paper, two test rig con gurationshence, withdifferent dynamic behaviorswere simulated showing theinstability present in each case. Te models consist o ashaf FEM with linear coefficients o stiffness and dampingrepresenting the journal bearings. Te different conditions o instability are compared through Campbell diagrams, orbits,and run-down simulations. Experimental evidences o oil-whirl and oil-whip are vastly discussed in the literature,mainly by the works o Muszynska [ , ]. Te experimentalmeasurements which motivated this work are also presentedand compared with the simulated results.

Tere ore, the investigation o the uid-induced instabil-ity in different conditions (here obtained through differenttest rigcon gurations)and itsdetectability in thedesignstageare the main contribution o this paper.

2. Materials and Methods

wo test rig con gurations were simulated to compare thedifferent conditions o instability; the nite element models(FEM) used are presented in Figure , and details o bothcon gurations are ound in able .

Con guration A (Figure (a)) is the same rom the workso Castro et al. [ ] and Mendes et al. [ ]. It contains a shaf


3/18


Coupling Bearing 1Disc

Shaf journal Bearing 2

1 3 11 16 19 21

(a)

Coupling Bearing 1 Disc Bearing 2 Shaf journal

1 4 10 16 21 23 24

Ballbearing

(b)

F : Finite element models: (a) con guration A; (b) con guration B.

: Details concerning both test r ig con gurations.

Shaf material Steel SAE Shaf diameter

mmShaf length mm (A) mm (B)Disc material Steel SAE Disc diameter mmDisc width . mmShaf journal material Steel SAE Shaf journal diameter mmShaf journal width mm

Coupling coefficients = = 1e N/m and = = 0N/mBall bearing coefficients = = 1e N/m and = = 0N/mJournal bearings radial clearance mJournal bearings diameter mmJournal bearings width mmLubricant ISO VG oil

supported by two hydrodynamic bearings symmetrically placed mm rom each other (nodes and ). It alsocontainsa discplacedat the shafmid span(nodes to ) anda journal at mm o the rst bearing (nodes to ). Teconnection between the end o the shaf and the WEG motoris made through a coupling (node ) modeled by constantlinear stiffness coefficients.

Con guration B (Figure (b)) consists o a shaf sup-ported by three bearings. wo journal bearings are placed

mm rom each other (nodes and ) and mm romthe right end o the shaf, where there is the motor coupling(node ). Te third bearing is a ball bearing at mm

rom the second journal bearing (modeled by constant linearstiffness coefficients at node ). In this con guration, thedisc lays between the journal bearings (nodes to ) and theshaf journal at mm be ore the ball bearing (nodes to

).As itcan be seenin Figure , the FEM model o con gura-

tion A was discretized in nodes; while con guration B was

F : Model o the journal bearings.

discretizedin nodes.Te position o the journal bearings ismarkedby black triangles, theballbearing by a white triangle,and the coupling by a grey triangle.

Te FEM matrixes used in the exible rotor models werethe classic Euler-Bernoulli beam matrixes or rotating shafs(i.e., including gyroscopic effect), as presented in Nelsonand McVaugh [ ] and here reproduced in the appendix.Te equation o motion is given in ( ), where is therotational speed, M is the mass matrix, G is the gyroscopicmatrix, K is the stiffness matrix, and C is the damping matrix(proportional to K by a actor o 2 104 according toSantana et al. [ ]). Besides, the matrix M is obtained romthe sum o two matrices ( ): the rotational mass matrix M Rand the translational mass matrix MT . F( ) is the external

orces vector, in this case, due to an unbalanced mass; q isthe generalized coordinate vector, with two translational andtwo rotational degrees o reedom (d.o. .) or each node:

Mq () + (C G)q () +Kq () = F(), ( )M = M T +M R. ( )

Te journal bearings are modeled by equivalent coeffi-cients o stiffness and damping (Figure ), added to the cor-responding d.o. . at the bearings location in the stiffness and


4/18


damping matrixes o the system, K and C , respectively. Tespeed dependent coefficients are evaluated rom Machado[ , , ] by the solution o the Reynolds equation ( ) using

nite volume method:

3 + 3 = 6 + 12. ( )In ( ), is axial coordinate, is the circum erential coor-dinate, is the viscosity o the uid, is the uid lm thick-

ness, is the pressure, and is the time. From the uid lmthickness, both the linearcoefficients (damping andstiffness)arecalculated through theperturbation method. Te pressuredistribution is evaluated rom Reynolds equation dependingon the uid lm thickness expression. Te reaction orces o the bearing can be obtained rom the integration o pressuredistribution. Tese orces are unctions o the displacementsand the instantaneous journal center velocities in directionand . Initially, the equilibrium position ( 0, 0) o the shafinside thebearing is ounddependingon therotational speed.

In this step, the term/is not considered. Aferwards, thecomplete Reynolds equation ( ) is solved again consideringsmall deviations around the equilibrium position, and. At this point, a aylor series expansion yields to ( ),

where the coefficients are partial derivatives evaluated at theequilibrium position ( 0, 0) as given by ( ):

= 0 + + + + , = 0 + + + + , ( ) =

0,

= 0. ( )

Lund [ ] pointed out that the solution o the Reynoldsequation should be solved simultaneously with the dynamicequation o the rotor; still, or practical purposes it can beassumed that thevibration amplitudeis small enoughtoallow substituting the uid lm orcesby their gradients around thesteady-state eccentricity or a given rotational speed; that is,the orcesbecome proportional to the displacement (stiffnesscoefficients) and velocity (damping coefficients). Lund [ ]also states that, although the coefficients de nition is only valid or in nitesimal amplitudes, comparisons with exactsolutions have shown that the coefficients approach may be applied or amplitudes as large as nearly orty percento the bearing clearance. It is also recommended that theeccentricity ratio lies below . - . .

In this work, the simulations per ormed presented muchlower amplitude levels than the limit stated by Lund [ ],except when the stability limit is reached. It is remarked herethatthe interesto the analysisper ormed is in ndingout theinstability threshold o the system and not the rotor behavioronce it is unstable.

Te coefficients behavior dependence on the rotationalspeed is shown in Figure (stiffness coefficients) andFigure (damping coefficients). Te eccentricities o the journal bearings used in both con gurations are presented in

Figure , which shows that below Hz, the bearings operatewith eccentricity ratio less than . , justi ying the linearizedcoefficients behavior in this range.

It should be noticed that, although the coefficients arespeed dependent, the model o the entire system (rotor andbearings) is linear, what implies that the system thresholds o

instability when per orming run-up and run-down simula-tions are the same.It is important to point out that the observance o the

instability threshold in a standard rotordynamic analysis( requency domain) is quite interesting rom the practicalpoint o view, being here the main ocus o this paper.

3. Results and Discussion

As stated be ore, in journal bearings supported rotors, twokinds o orward sel -excited vibrations are present: oil-whirland oil-whip [ ]. As vastly discussed in the literature, oil-whip is an unstable vibrationwhich occursnear twice the rst

critical speed o the rotor, occurring whenoil-whirl coincideswith the natural requency o the shaf. Tis phenomenonwas simulated and experimentally observed in a test rigused in previous works [ , ], here called con gurationA. However, when the test rig con guration was changed(con guration B) a different kind o instability was observedin the simulations and in the experimental setup. In orderto better understand the different instability mechanisms, aseries o simulations was proposed, which is presented in thissurvey.

. . Homogeneous Solution. From the solution o the homo-geneous part o ( ) it is possible to obtain the Campbelland modal damping diagrams and the natural modes o thesystem. Figure presents the Campbell and modal dampingdiagrams or both con gurations. In the Campbell diagram,when the rotational speed curve ( x line) crosses one o thenatural requencies o the rotor, there is a resonance regionand the rotational speed, at this point, is called critical speed.Hence, a critical speed is clearly observed in con guration A(Figure (a)) at approximately . Hz while in con gurationB (Figure (c)) close to Hz. When simulating a shaf niteelementmodelcontaining journal bearings modeled by linearcoefficients o stiffness and damping, two natural requencies(one or each bearing) can be ound near hal the rotationalspeed o the rotor (Figures (a) and (c)).

In act, these requencies show up due to the effect o the journal bearings stiffness coefficients representing the uid

lm.Te presenceo such requenciesallows the linearmodelto represent the uid-induced vibrations phenomena witha very good agreement with experimental observations o rotor dynamic behaviors. Muszynska [ ] simulated a exiblerotorsupported byonerigid bearing andonejournal bearing,modeled by the modal parameters o the rst bending mode.Tree eigenvalueswereencountered,with twoo them relatedto the rst bending modes and one with the requency (imaginary part) close to the oil-whirl requency. Te authornoted that the real part o this third eigenvalue predicts thethreshold o instability. In the words o Muszynska [ ], this


5/18


10 20 30 40 50

2

1.5

1

0.5

0

0.5

1

1.5

106

Rotational speed (Hz)

S t i ff n e s s c o e ffi c i e n

t s ( N / m )

(a)

10 20 30 40 50

2

1.5

1

0.5

0

0.5

1

2

1.5

106


S t i ff n e s s c o e ffi c i e n

t s ( N / m )

(b)

20 40 60 80 100 120 140

5

4

3

2

1

0

1

2

3

4106


S t i ff n e s s c o e ffi c i e n t s

( N / m

)

KyyKyz

KzyKzz

(c)

20 40 60 80 100 120 140

5

4

3

2

1

0

1

2

3

4106


S t i ff n e s s c o e ffi c i e n t s

( N / m

)

KyyKyz

KzyKzz

(d)

F : Equivalent coefficients o stiffness or the journal bearings: (a) bearing in con guration A; (b) bearing in con guration A; (c)bearing in con guration B; (d) bearing in con guration B.

is an unconventional uid/solid interaction-related naturalrequency o the rotor/bearing system, hereinafer denoted

as uid frequencies. As it was shown, in the nite elementmodels o the rotors considered in this paper, there were two

uid requencies, one related to each journal bearing.

Moreover, Figure (a) shows one o the uid requenciescrossing the rst natural requency o the rotor when therotational speed is o approximately . Hz, what meansto excite the rst mode o the rotor at a rotational speednearly twice the critical speed o the rotor. It can also be


6/18


10 20 30 40 50

5

0

5

10

15

20

104


D a m

p i n g c o e ffi c i e n

t s ( N

s / m

)

(a)

0.5

0

0.5

1

1.5

2

2.5

3

105

10 20 30 40 50Rotational speed (Hz)

D a m

p i n g c o e ffi c i e n

t s ( N

s / m

)

(b)

20 40 60 80 100 120 140

0

5

10

15

104


D a m

p i n g c o e

ffi c i e n

t s ( N

s / m

)

CyyCyz

CzyCzz

(c)

0

1

2

3

20 40 60 80 100 120 140

105


D a m

p i n g c o e

ffi c i e n

t s ( N

s / m

)

CyyCyz

CzyCzz

(d)

F : Equivalent coefficients o damping or the journal bearings: (a) bearing in con guration A; (b) bearing in con guration A; (c)bearing in con guration B; (d) bearing in con guration B.

seen, in the modal damping diagram o con guration A(Figure (b)), thatneartwice the rst natural requencyo therotor ( . Hz) one o the uid requencies damping assumesa negative value, meaning that the system became unstableafer the requency o Hz. Tis instability threshold occurs very close to the oil-whip requency, as it can be seen in

the Campbell diagram. Once this instability threshold isachieved, its requencydoes notchange anymore, because therotor enters in the oil-whip instability and it develops a highlevel vibration with a requency component near its natural

requency, which does notchange even i the rotoraccelerates[ ]. During oil-whip the bearing vibration amplitude is


7/18


10 20 30 40 500

0.2

0.4

0.6

0.8

1


E c c e n

t r i c i t y

Bearing 1ABearing 2A

(a)

50 100 1500

0.2

0.4

0.6

0.8

1


E c c e n

t r i c i t y

Bearing 1BBearing 2B

(b)

F : Journal bearings eccentricity: (a) con guration A; (b) con guration B.

limited by the bearing clearance, but the rotor develops amuch higher vibration level, once its natural requency isbeing excited, as previously mentioned.

Analogously, Figure (c) shows the uid requency cross-ing the rst natural requency ( Hz) at the rotational speedo Hz, which characterizes an oil-whip requency in thiscase. However, the modal damping diagram o con gurationB (Figure (d)) shows that this con guration has an instabil-itythreshold at Hz,which is much lower than theexpectedoil-whip instability, what suggests that a different kind o

uid-induced instability occurs. Te behavior o the uidrequency that gets unstable didnotbecome constant as it did

in con guration A, pointing to the oil-whirl as the instability cause, although it is interesting to point out that near theexpected oil-whip requency ( Hz) the tendency o beingconstant is observable, suggesting that oil-whip instability would happen at that requency i the rotor did not getunstable sooner or another reason.

Te mode shapes o con guration A are shown inFigure , and in Figure , or con guration B, respectively.Figures (a)and (b) give the natural mode o the rotor at therotational speed o Hz, corresponding respectively to the

uid requencies o . Hz and . Hzregarding to thehydrodynamic bearings (whirl precession). Figures (c) and

(d) bring the orward ( . Hz) and backward ( . Hz)natural modes o therotor when therotational speed is Hz.

Te mode shapes o the uid requencies or con gura-tion B (Figures (a) and (b)) show that each uid requency corresponds to each o the journal bearings due to the actthat in each mode shape one o the bearings presents a high

displacement; such effect is not so clear in con guration A(Figure (a)).

Up to this point, it was shown that through the homo-geneous solution o the system equation, con guration Apresents oil-whip instability, but con guration B presentsan instability threshold lower than expected, suggesting thatoil-whirl dominates the unbalance response, de ning thethreshold o instability.

. . Forced Response. In order to obtain a better understand-ing o the instability mechanism, run-down simulations wereper ormed or both con gurations (Figure ). Con gurationA was simulated rom Hzto Hz, once the analysisshowedthe system became unstable at Hz, and con guration B

rom Hz to Hz, because the modal damping diagram(Figure (d)) indicates that instability starts at Hz, but theCampbell has evidence that oil-whip would happen nearly at

Hz. In both cases, a deceleration o Hz/s was used.

Remark. Te reason why run-down simulations were usedinstead o run-ups is related to the instability sensitivityo thelinear model used or the journal bearings. It was noticed inrun-up simulations that, even though the rotor was unstableafer the instability threshold, the vibration amplitude risesin a very slow shape. Consequently, a run-down simulationmay be set to start in a rotor unstable condition, with large vibrationamplitudes,overcomingthe run-up simulation lim-itation.Hence, run-downsimulations aresuitable to thestudy o uid-induced instability simulations in cases where, likein this work, the entire system model is linear (recalling that


8/18


0 26.6 40 49 540

10

20

30

40

50

60


w n

( H z )

(a)

0 20 40 60

0

0.2

0.4

0.6

0.8


D a m

p i n g r a

t i o

( q s i )

48.8

(b)

0 40 75 100.1 1500

50

100

150


w n

( H z )

1st uid2nd uid

1st FW1st BW

(c)

0 50 100.1 150

0

0.2

0.4

0.6

0.8

1


D a m

p i n g r a

t i o

( q s i

)

1st uid2nd uid

1st FW1st BW

(d)

F : (a) Campbell diagram o con guration A; (b) modal damping o con guration A; (c) Campbell diagram o con guration B; (d)modal damping o con guration B.

the journal bearings were modeled using a linear coefficientsapproach), and the same instability thresholds are obtainedeither or run-up or run-down simulations. It should also benoticed that the journal bearings nodes vibration is limitedby the bearing radial clearance. Te interaction between the journal and the housing wall is not in the scope o this paper.Tereby, a simple approach was used: when the vibrationamplitude gets higher than the radial clearance, the angular

position o the shaf inside the bearing is kept constantand itsamplitude is reduced to theclearancevalue.Te disadvantageo this approach is that harmonic components showed upin the water all diagrams. Hence, their in uence should bedisregarded in the analysis.

Figures (a) and (c) show the time response or the run-down in the rst journal bearing o each con guration, whereit is clear that the systems were unstable once the stability


9/18


0

0.5

1

0

1

1

0.5

0

0.5

1

103

103

xy

z

(a)

0

0.5

1

0

1

1

0

1

103

103

xy

z

(b)

0

0.5

5

0

5

5

0

5

xy

z

104

104

(c)

0

0.5

5

0

5

5

0

5

xy

z

104

104

(d)

F : Natural modes o con guration A or a rotational speed o Hz: (a) bearing uid requency at . Hz; (b) bearing uidrequency at . Hz; (c) rst orward exural natural requency at . Hz; (d) rst backward exural natural requency at . Hz.

thresholds were crossed. Te sel -centering effect o the shafdue to unbalance (reduction o the orbit amplitude afer thecritical speed) and the sel -centering effect o the bearing canalso be noticed.

Figures (b) and (d) contain the water all diagramscorrespondent to Figures (a) and (c), respectively. Oil-whirl did not show up in con guration A (Figure (b)). Tesystem presents only the unbalance response until oil-whiptakes place, when the modal damping o the uid requency crosses the zero. In real rotors, it can be seen that the orbito the shaf inside the bearings presents a deployment due toa subharmonic component at hal the rotational speed (i.e.,oil-whirl); and when the rotor rotational speed has a smallincrement and crosses the instability threshold, the oil-whipbecomes dominant [ ]. Figure presents the orbits and

the spectra o the rst bearing in several constant rotationalspeeds (dashed lines in Figure (a)), be ore the critical speed( Hz), at the critical speed ( . Hz), between the criticalspeed and the instability threshold ( Hz), and at . Hz(close to the instability), and no deployment can be seen,which demonstrate a limitation o the linear model oncethis phenomenon was observed in the nonlinear model usedin the work o Castro et al. [ ], although the instability threshold presented good agreement. Te large orbit seen inFigure (a) at Hz is a transient effect o the system goingout o the instability region (near . Hz), once this resultwas obtained rom a run-down simulation as previously mentioned.

In the case o con guration B, the time response o therun-down (Figure (c)) alsoshows the instability o the rotor,


10/18


0

0.5

1

0

1

1

0

1

xy

z

103

103

(a)

0

0.5

5

0

5

5

0

5

xy

z

104

104

(b)

0

0.5

1

01

1

0

1

xy

z

104

104

(c)

0

0.5

1

01

1

0

1

xy

z

104

104

(d)

F : Natural modes o con guration B or a rotational speed o Hz: (a) bearing uid requency at . Hz; (b) bearing uidrequency at . Hz; (c) rst orward exural natural requency at . Hz; (d) rst backward exural natural requency at . Hz.

when the orbit amplitude is limited by the bearing clearance.However, in the water all diagram it can be seen that nearthe instability threshold ( Hz) oil-whirl rises and starts tocausean unstable vibration. Itis possible to observe it becauseas the rotational speed changes, the unstable harmoniccomponent o the vibration also changes, being close to %o the rotational speed until nearby Hz, when oil-whipis expected to occur even though the system was already unstable because o oil-whirl. Tereby, in con guration B,instead o oil-whip, oil-whirl was the responsible or drivingthe system unstable.

Constant rotational speed simulations were also per-ormed or two cases in con guration B, at Hz and close

to the instability limit at Hz; the orbits o the rst bearingand the requency spectrum are presented in Figure . Teexperimental results, which have motivated this survey, are

also shown in Figure overlaid to the simulated results tohighlight the differences. All measurements were made using

RAM inductive sensors rom Bently Nevada.At the rotational speed o Hz (gray curves in

Figures (a) and (b)) only the unbalance effect appears,but as the rotational speed was increased to Hz, theorbit deployment appears (Figure (c) in gray), indicatingthe presence o oil-whirl as it can be seen in the requency spectrum (Figure (d) in gray).

Te experimental threshold o instability is slightly shif-ed to . Hz. Tere ore, the rst difference between exper-iment and simulation is in the constant rotational speedsmeasured, . Hzand . Hz,whichshows that themodelis a littlebit more conservative. Te orbit shapes (blackcurvesin Figures (a) and (c)) are considerably different rom thesimulated results, and the reason is the presence o many


11/18


5 10 15 20 25 30 35 40 45 505

0

5

y ( m )

5

0

5

Frequency (Hz)

Displacement-direction y

5 10 15 20 25 30 35 40 45 50

z ( m )

Frequency (Hz)

Displacement-direction z105

105

(a)

105

M a g n i

t u d e ( m ) 3

2

1

010

10

202030

30

40

40

50

50

F r e q u e n c y ( H z ) R o t

a t i o n ( H

z )

1x

(b)

20 40 60 80 100 120 140

5

0

5

y ( m )

Frequency (Hz)

20 40 60 80 100 120 140Frequency (Hz)

Displacement-direction y105

5

0

5

105

z ( m )

Displacement-direction z

(c)

20 40 60 80 100 120 14050

100

1500

1

2

3

4

105

1x

R o t a t

i o n ( H

z )

0.46x

F r e q u e n c y ( H z )

M a g n i

t u d e ( m )

(d)

F : Run-down simulations: (a) time response o the rst bearing in con guration A (see Figure ); (b) water all diagram o the rstbearing in con guration A; (c) time response o the rst bearing in con guration B; (d) water all diagram o the rst bearing in con gurationB.

harmonics ound in the experimental requency spectra(Figures (b) and (d) in black). Such harmonics occur dueto the ball bearing at the shaf end in con guration B; as therotor spins and its spheres pass through the lower part o thebearing, where theload isapplied,harmonics o therotationalspeed are excited and participate in the vibration response.Tis effect is not present in the simulations once the ballbearing is merely modeled by constant stiffness and dampingcoefficients (Figure (b)). However, the rotational speedsassociated with the threshold o instability are very closeand the oil-whirl components in the spectra present aboutthe same amplitudes obtained by the simulation response(Figure (d)).

Tere ore, comparing the two different cases, it can beseen that in one case the instability was caused by oil-whirl

and, in the other case, by oil-whip. It should be remindedat this point that oil-whirl could exist in a stable conditionuntil it turns to oil-whip, as presented in Castro et al. [ ];such behavior would be expected by a lighter rotor than theones used in the present survey. Moreover, it was shown herethat the instability threshold could be sa ely estimated whenusing the linear coefficients approach or journal bearingsmodeling.

. . Eccentricity Analysis. Te denominations heavy rotor or light rotor used when talking about horizontal rotorssupported on journal bearings are related to the eccentricity o the shaf during operation. Te bearing eccentricity is a

unction o the load applied on each bearing and the bearing


12/18


2.625 2.63 2.635

3.565

3.56

3.555

3.55

z ( m )

105

105y (m)

(a)

10 20 30 40 500

0.5

1

1.5

2

2.5

3

107

A m

p l i t u

d e ( m

)

Frequency (Hz)

(b)

2.2 2.3 2.4 2.5

105

105

1.45

1.4

1.35

1.3

1.25

1.2

1.15

z ( m )

y (m)

(c)

10 20 30 40 500

0.5

1

1.5

2106

A m

p l i t u

d e ( m )

Frequency (Hz)

(d)

1.94 1.96 1.98 2 2.02

106

1058.8

8.6

8.4

8.2

8

z ( m

)

y (m)

(e)

10 20 30 40 500

0.5

1

1.5

2

2.5

3

107

A m

p l i t u

d e ( m )

Frequency (Hz)

( )

F : Continued.


13/18


1.53 1.54 1.55 1.56 1.57

105

5.1

5

4.9

4.8

105y (m)

z ( m )

(g)

10 20 30 40 500

0.5

1

1.5

2

2.5

3107

Frequency (Hz)

A m

p l i t u

d e ( m

)

(h)

F : Simulated results o the rst bearing in con guration A or constant rotational speeds (see Figure (a)): (a) orbit at Hz; (b)requency spectrum at Hz; (c) orbit at Hz; (d) requency spectrum at Hz; (e) orbit at Hz; ( ) requency spectrum at Hz; (g) orbit

at Hz; (h) requency spectrum at Hz.

load capacity at each rotational speed. In this sense, theterminology heavy rotor and light rotor re ers to the relationbetween bearing load and load bearing capacity and how itaffects the bearing eccentricity. Te bearing load is constantand dependenton thegeometry o the rotating system; yet, asbearing load capacity is a unction o the rotational speed, sois the bearing eccentricity.

When operating at low rotational speeds, a journal bear-ing pumps a little quantity o oil underneath its journal,

resulting in a thin stiff uid- lm and in a high eccentricity;thus, oil-whirl is not present and the rotor is stable.As the rotational speed increases, the decreasing in the

eccentricity and the rising o the attitude angle (tending to degrees) results in the sel -centering effect. Te thick uid

lm causes disequilibrium o the stabilizing orces inside thebearings; the same condition o low eccentricity occurs in vertical or lightly loaded horizontal rotors as they operatenearly centered.

Figure presents the orces acting on the shaf inside thebearing as terms o the stiffness and damping coefficients.In order to maintain stable operation, the dissipative orce( and ) must suppress the destabilizing effect o thecross-coupled stiffness ( and ). However, as the rota-tional speed increases, the cross-coupled stiffness coefficientsincrease drastically, while thedirectstiffness and thedampingcoefficients present only small changes (Figures and ),which causes a reduction o stability and oil-whirl may appear.

I the rotor keeps on accelerating, it will achieve theinstability threshold and oil-whirl may become unstable (asin con guration B) or it may give place to oil-whip (as incon guration A).

Consequently, the different behaviors o the instability mechanism in both con gurations A and B are due to therelation between bearing load and bearing load capacity.

I therotor hasone bearing working in a light rotor condition,such that it became unstable be ore oil-whirl reaches thenatural requency o the rotor (and turn to oil-whip), thenoil-whirl will be the cause o the instability. I the bearingswork in a heavy rotor condition (higher eccentricity), thenoil-whip will cause the instability at twice the rotor natural

requency. Oil-whirl instability threshold here is consideredtobe thepointwhere thesubsynchronous harmonic is limitedonly by the bearing clearance; and this effect is observed in

the linearmodelsas a negativevalue associated with the uid-requency in the modal diagrams.Since the same bearings were used in both cases, the load

in each case tellswhich bearingis more unstable.Te loads orcon guration A are . N or the rst bearing and . N orthe second, and con gurationB has . N on the rstbearingand . N on the second. Tis means that the rotor romcon guration B has a bearing with a much lower eccentricity (less stable) as it can be seen in Figure , what explains why the instability mechanism has a different behavior rom theprevious con guration. In con guration B, oil-whirl becameunstable much sooner than the expected oil-whip.

Only or comparison, the load in the ball bearing in

con guration B is . N; there ore, the sum o the loadso the bearings in each con guration allows observing thatboth con gurations have almost the same weight ( . N

or con guration A and . N or con guration B). Tisact agrees with the early explanation between how the rotor

weight is not the only parameter related to the light/heavy rotor denominations, but it strongly depends on the loadapplied to each bearing and the bearing load capacity.

4. Conclusions

Motivated by experimental results, a survey about the uid-induced instability mechanism was presented considering


14/18


4 2 0 2

105

3

2

1

0

1

2

3

4

105

y (m)

z ( m )

(a)

100 200 300 400 5000

0.5

1

1.5

2

2.5105

Frequency (Hz)

A m

p l i t u

d e ( m )

(b)

5 0 56

4

2

0

2

4

6

8

105

105y (m)

z ( m )

ExperimentalSimulated

(c)

100 200 300 400 5000

1

2

3

4

5

6105

Frequency (Hz)

A m p l i t u

d e ( m )

ExperimentalSimulated

(d)

F : Experimental and simulated results or two conditions with xed rotational speeds in con guration B: (a) orbits o the rstcondition (experimental Hz, simulated at Hz); (b) requency spectrum o the rst condition (experimental at Hz, simulated at

Hz); (c) orbits o the second condition (experimental at . Hz, simulated at . Hz); (d) requency spectrum o the second condition(experimental at . Hz, simulated at . Hz).

two different rotor con gurations. Te hydrodynamic bear-ings were modeled as linear coefficients o stiffness anddamping, evaluated through the solution o the Reynoldsequation using a nite-volume method. Te rst con gura-tion instability threshold is the expected oil-whip requency;

however, in the second con guration, the instability thresh-old occurs much sooner than expected, being caused by anunstable oil-whirl vibration. Te analysis and comparison o both cases were made through Campbell andmodal dampingdiagrams and, aferwards, run-down simulations (water all


15/18


Fdy

Z

z

zFsyzFsz Fszy

y

Fsy

Fdz

Y

y

Fsyz = Kyz z

Fsz = Kzz z

Fdy = Cyy y

Fszy = Kzy y

Fsy = Kyy y

Fdz = Czz z

Whirldirection

F :Representation o the orces actingon theshaf inside the journal bearing.

diagram) validation. In both cases, the instability thresh-old and the instability cause (oil-whirl or oil-whip) couldbe success ully identi ed through the models, presentinggood agreement with experimental results even though no

model adjustment techniques were used. Hence, the linearcoefficients approach could be used in the design stage asa rst indicative o the machine instability limit. Finally, adiscussion about the instability mechanism and how it isaffected by thebearing load andthebearing load capacity waspresented.

Appendix Te matrixes used to build the nite element model are givenby

M T = 420

1560 156 sym.0 22 42

22 0 0 4254 0 0 13 1560 54 13 0 0 1560 13 32 0 0 22 42

13 0 0 32 22 0 0 42

,

M R = 2480

360 36 sym.0 3 4

2

3 0 0 4236 0 0 3 360 36 3 0 0 360 3 2 0 0 3 423 0 0 2 3 0 0 42

,

G = 2240

036 0 skew sym.3 0 00 3 42 00 36 3 0 036 0 0 3 36 03 0 0 2 3 0 00 3 2 0 0 3 42 0

,

K = 3

120 12 sym.0 6 426 0 0 4212 0 0 6 120 12 6 0 0 120 6 22 0 0 6 426 0 0 22 6 0 0 42

,

(A. )

where the area moment o inertia is given by = = 644. (A. )

Notation

Roman Letters

: Shaf element diameter

: Fluid lm thickness: Shaf element length

: Pressure


16/18


q: Generalized coordinate vector: ime,: Axial and circum erential coordinates o the

bearing,: Coordinates o the plane perpendicular tothe shaf

: Cross-section area o the shaf element: Damping coefficientC: Damping matrix

: Young modulusF: Force vectorG: Gyroscopic matrix, : Area moments o inertia

: Stiffness coefficientK: Stiffness matrixM: Mass matrixMT : ranslational mass matrixMR : Rotational mass matrix.

Geek Letters: In nitesimal variation: Shaf material density : Fluid viscosity : Rotational speed.

Superscripts

: Differentiation with respect to time: Force due to a damping coefficient: Force due to a stiffness coefficient.

Subscript : Equilibrium position.

Conflict of Interests

Te authors declare that there is no con ict o interestsregarding the publication o this paper.

Acknowledgments

Teauthorswould like to thank the unding support byCNPq(Conselho Nacional de Desenvolvimento Cient co e ec-

nologico) (Grant no. / - ); CAPES (Coordenacaode Aper eicoamento de Pessoal de Nvel Superior) (PRO-BRAL / ); and FAPESP (Fundacao de Amparo a Pesquisado Estado de Sao Paulo) (Grant no. / - ).

References

[ ] N. P. Petrov, Friction in machines and the effect o lubricant,Inzhenernyj Zhurnal, Sankt-Peterburg , vol. , pp. , .




[ ] B. ower, First report on riction experiments ( riction o lubri-cated bearings), pp. , , (Adjourned Discussion, pp.

, ).[ ] B. ower, Second report on riction experiments (experiments

on the oil pressure bearing), Proceedings of the Institution of Mechanical Engineers, pp. , .[ ] O. Reynolds, On thetheoryo lubrication andits application to

Mr. Beauchamptowers experiments, includingan experimentaldetermination o the viscosity o olive oil, Philosophical rans-actions of the Royal Society A, vol. , no. , pp. , .

[ ] G. Oliver, An Introduction to Oil Whirl and Oil Whip, urboComponents and Engineering Newsletter: Bearing Journal , vol. ,no. , pp. , .

[ ] B. L. Newkirk, Shaf Whipping, General Electric Review, vol., article , .

[ ] B.L. Newkirk andH. D. aylor, Shafwhippingdue tooil actionin journal bearings, General Electric Review, vol. , pp.

, .[ ] A. Muszynska, Whirl and whiprotor/bearing stability prob-

lems, Journal of Sound and Vibration , vol. , no. , pp. , .

[ ] A. Muszynska, Stability o whirl and whip in rotor/bearingsystems, Journal of Sound and Vibration , vol. , no. , pp.

, .[ ] A. Stodola,Kritische Wellenstorungin olgeder Nachgiebigkeit

desOelpolsters im Lager, SchweizerischeBauzeitung ,vol. - ,pp. , (German).

[ ] C. Hummel, Kritische drehzahlen als folge der nachgiebigkeit desschmiermittels im lager [Tesis] , Eidgenossischen echnischenHochschule, Zurich, Switzerland, (German).

[ ] J. W. Lund, Spring and damping coefficients or the tilting pad journal bearing, ASLE ransactions, vol. , no. , pp. ,

.[ ] . H. Machado and K. L. Cavalca, Evaluation o dynamic

coefficients o uid journal bearings with different geometries,in Proceedings of the th Brazilian congress of mechanical engineering (COBEM ), ABCM, Gramado, Brazil, November

.[ ] . H. Machado, Evaluation of hydrodynamic bearings with

geometricdiscontinuities[Dissertation], Universityo Campinas,Campinas, Brazil, (Portuguese).

[ ] G. Capone, Orbital motions o rigid symmetric rotor sup-ported on journal bearings, La Meccanica Italiana, vol. , pp.

, (Italian).

[ ] G. Capone, Analytical description o uid-dynamic orce eldin cylindrical journal bearing, LEnergia Elettrica, vol. , pp.

, (Italian).[ ] J. M. Vance, Rotordynamics of urbomachinery , John Wiley &

Sons, New York, NY, USA, st edition, .[ ] E. Kramer, Dynamics of Rotors and Foundations, Springer, New

York, NY, USA, st edition, .[ ] D. Childs, urbomachinery Rotordynamics: Phenomena, Model-

ing, and Analysis, Wiley-Interscience, New York, NY, USA, stedition, .

[ ] E. N. Lima,Non-linearmodel for hydrodynamic sustaining forceson vertical rotors journal bearings [Dissertation] , University o Campinas, Campinas, Brazil, (Portuguese).


17/18


[ ] H. F. de Castro, K. L. Cavalca, and R. Nordmann, Whirland whip instabilities in rotor-bearing system considering anonlinear orce model, Journal of Sound and Vibration, vol. ,no. - , pp. , .

[ ] H. D. Nelson and J. M. McVaugh, Te dynamics o rotor-bearing systems using nite elements, Journal of Engineering for Industry , vol. , no. , pp. , .

[ ] A. A. G. Siqueira, R. Nicoletti, N. Norrick et al., Linear param-eter varying control design or rotating systems supported by journal bearings, Journal of Sound and Vibration, vol. , no.

, pp. , .[ ] B. Riemann, E. A. Perini, K. L. Cavalca, H. F. Castro, and S.

Rinderknecht, Oil whip instability control using -synthesistechnique on a magnetic actuator, Journal of Sound and Vibration , vol. , no. , pp. , .

[ ] J. uma, J. Simek, J. Skuta, and J. Los, Active vibrations controlo journal bearings with the use o piezoactuators, Mechanical Systems and Signal Processing , vol. , no. , pp. , .

[ ] R. U. Mendes, L. O. S. Ferreira, and K. L. Cavalca, Analysis o a complete model o rotating machinery excited by magnetic

actuator system, Proceedings of the Institution of Mechanical Engineers C , vol. , no. , pp. , .[ ] P. M. Santana, K. L. Cavalca, E. P. Okabe, and . H. Machado,

Complex response o a rotor-bearing- oundation system,in Proceedings of the th IF oMM International Conferenceon Rotordynamics, vol. , pp. , KAIS University, Seoul,Republic o Korea, September .

[ ] . H. Machado and K. L. Cavalca, Dynamic analysis o cylin-drical hydrodynamic bearings with geometric discontinuities,in Proceedings of the th International Conference on VibrationProblems (ICOVP ), J. Naprstek, J. Horacek, M. Okrouhl k,B. Marvalova, F. Verhulst, and J. . Sawicki, Eds., vol. o Proceedings in Physics, pp. , Springer, Prague, CzechRepublic, September .

[ ] J. W. Lund, Review o the concept o dynamic coefficients oruid lm journal bearings, Journal of ribology , vol. , no. ,

pp. , .


18/18

Submit your manuscripts athttp://www.hindawi.com

Documents

On the Instability Threshold of Journal Bearing Supported Rotors_VERY GOOD